Topological Quantization of Complex Velocity in Stochastic Spacetimes

Abstract

We establish a formal geometric framework for quantum fields on a stochastic gravitational background. Starting from a master partition function that averages over metric fluctuations, we define a matter amplitude $\mathcal{K}[\Phi ;x]$ with explicit parametric dependence on spacetime coordinates, whose logarithmic derivative yields a complex velocity field ${\eta }_{\mu }={\pi }_{\mu }-i{u}_{\mu }$. This object is shown to be a section of the pullback bundle $E={\pi }_2^{\ast }(T^{\ast }M)$ over the product of configuration space $\mathcal{C}$ and spacetime $M$, and defines a flat ${\mathbb{C}}^{\times }$ connection whose $U(1)$ part governs phase transport and whose ${\mathbb{R}}^{+}$ part governs scale transport. The coupled dynamics collapse into the complex geodesic equation ${\mathcal{L}}_{\eta }\eta =d(|\eta {|}^2)$. On non-simply connected spacetimes, the total phase satisfies $\frac{m}{\hslash }{\oint }_{\gamma }{\eta }_{\mu }d{x}^{\mu }=2\pi n+\Delta {\varphi }_{\mathrm{top}}$, demonstrated analytically for a scalar field with twisted boundary conditions on a conical spacetime. We provide a self-contained bundle isomorphism mapping ${\eta }_{\mu }$ to the symmetric logarithmic derivative of quantum estimation theory, and derive the associated Berry curvature. A prospective estimate for the MAGIS-100 interferometer yields ${\gamma }_B\sim 3.97\times 10^{-9}$ rad for an astrophysical stochastic background (approximately five orders below projected sensitivity) and ${\gamma }_B\sim 3.97\times 10^{-53}$ rad for Planck-scale fluctuations. This framework geometrizes quantum mechanics without hidden variables: stochasticity imprints spacetime fluctuations on matter, preserving the wave function's probabilistic nature while maintaining consistency with the Born rule.